Define a fibonacci series generated by {a,b} such as,

(The original fibonacci series is generated by {1,1})

( )

Then there are a number of interesting properties:

1)If is the finite continued fraction for

then

Where the appears times.

2)Thus, from here we have that

(Thus, any fibonacci series generated by any two real numbers converges to the divine proportion).

3)The formula for is given by

where is the n-th fibonacci number.

But by Binet's formula we have that,

for the n-th fibonacci number we can find a formula for but it is rather messy and will be omitted. Giving a second method for proving statement 2.